SDLS: a Matlab package for solving conic least-squares problems

This document is an introduction to the Matlab package SDLS (Semi-Definite Least-Squares) for solving least-squares problems over convex symmetric cones. The package is shortly presented through the addressed problem, a sketch of the implemented algorithm, the syntax and calling sequences, a simple numerical example and some more advanced features. The implemented method consists in solving the dual problem with a quasi-Newton algorithm. We note that SDLS is not the most competitive implementation of this algorithm: efficient, robust, commercial implementations are available (contact the authors). Our main goal with this Matlab SDLS package is to provide a simple, user-friendly software for solving and experimenting with semidefinite least-squares problems. Up to our knowledge, no such freeware exists at this date. 1 General presentation SDLS is a Matlab freeware solving approximately convex conic-constrained least-squares problems. Geometrically, such problems amount to finding the best approximation of a point in the intersection of a convex symmetric cone with an affine subspace. In mathematical terms, those problems can be cast as follows: minx 1 2 ‖x − c‖2 s.t. Ax = b x ∈ K (1) where x ∈ R has to be found, A ∈ R, b ∈ R, c ∈ R are given data and K is a convex symmetric cone. The norm appearing in the objective function is ‖x‖ = √ xT x, the Euclidean norm associated with the standard inner product in R. Note that there exists a unique solution to this optimization problem. In practice, K must be expressed as a direct product of linear, quadratic and semidefinite cones. It is always assumed that matrix A has full row-rank, otherwise the problem can be reformulated by eliminating redundant equality constraints. LAAS-CNRS, University of Toulouse (France) Czech Technical University in Prague (Czech Republic) CNRS, Laboratoire Jean Kunztmann, University of Grenoble (France)